route inspection problemtitle route inspection problem author a g martin subject discrete...
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0.1
The Route Inspection ProblemThe Route Inspection Problem
also known as
The Chinese Postman ProblemThe Chinese Postman Problem
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0.2
Node type
An n-node is a node where n arcs join
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0.2
Node type
An n-node is a node where n arcs join and is
said to be a node of order n.
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0.2
Node type
An n-node is a node where n arcs join and is
said to be a node of order n.
1-nodes, 3-nodes, (and so on) are called odd
nodes or nodes of odd order.
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0.2
Node type
An n-node is a node where n arcs join and is
said to be a node of order n.
1-nodes, 3-nodes, (and so on) are called odd
nodes or nodes of odd order.
2-nodes, 4-nodes, (and so on) are called even
nodes or nodes of even order.
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0.3
Numbers of nodes
In any network there is always an even number
of odd nodes.
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0.3
Numbers of nodes
In any network there is always an even number
of odd nodes.
Add together the node type numbers of all the
nodes in a network.
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0.3
Numbers of nodes
In any network there is always an even number
of odd nodes.
Add together the node type numbers of all the
nodes in a network. The sum will be twice the
number of arcs because each arc is counted
twice, once from each end.
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0.3
Numbers of nodes
In any network there is always an even number
of odd nodes.
Add together the node type numbers of all the
nodes in a network. The sum will be twice the
number of arcs because each arc is counted
twice, once from each end.Thus the sum of
node type numbers is even for any network.
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0.4
An odd number of odd nodes cannot exist
because a sum containing an odd number of
odd numbers would be odd. Thus any network
must have an even number of odd nodes (if it
has any odd nodes).
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0.5
Traversability
A network is said to be traversable (or
Eulerian) if you can draw it without removing
your pen from the paper and without retracing
the same arc twice.
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0.5
Traversability
A network is said to be traversable (or
Eulerian) if you can draw it without removing
your pen from the paper and without retracing
the same arc twice. For a network to be
traversable it must have 0 or 2 odd nodes
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0.5
Traversability
A network is said to be traversable (or
Eulerian) if you can draw it without removing
your pen from the paper and without retracing
the same arc twice. For a network to be
traversable it must have 0 or 2 odd nodes and if
we are able to start and finish at the same
node it must have no odd nodes.
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0.6
Chinese Postman ProblemChinese Postman Problem
The Chinese postman problem concerns a
postman who has to deliver mail to houses
along each of the streets in a particular
housing estate,
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0.6
Chinese Postman ProblemChinese Postman Problem
The Chinese postman problem concerns a
postman who has to deliver mail to houses
along each of the streets in a particular
housing estate, and wants to minimse the
distance he has to walk.
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0.6
Chinese Postman ProblemChinese Postman Problem
The Chinese postman problem concerns a
postman who has to deliver mail to houses
along each of the streets in a particular
housing estate, and wants to minimse the
distance he has to walk.
The problem was first considered by the
chinese mathematician Mei-ko Kwan in the
1960s, hence its name.
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0.7
How does this problem differ from the problem
of the travelling salesman?
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0.7
How does this problem differ from the problem
of the travelling salesman?
The travelling salesman needs to visit every
node (or vertex) whereas the Postman needs
to travel along each arc (or edge).
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0.7
How does this problem differ from the problem
of the travelling salesman?
The travelling salesman needs to visit every
node (or vertex) whereas the Postman needs
to travel along each arc (or edge).
Therefore it is required that the network is
traversable (Eulerian).
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0.8
The problem is therefore to obtain “a closed
trail of minimum weight containing every arc”.
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0.8
The problem is therefore to obtain “a closed
trail of minimum weight containing every arc”.
Or put another way, “find a route of minimum
weight which traverses every edge at least
once, returning to the start vertex”.
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0.8
The problem is therefore to obtain “a closed
trail of minimum weight containing every arc”.
Or put another way, “find a route of minimum
weight which traverses every edge at least
once, returning to the start vertex”.
We first deal with traversability (the closed
trail), then consider the mimimum weight
condition (shortest distance).
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0.9
Recall the result about traversability
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0.9
Recall the result about traversability
For a network to be traversable it must have 0
or 2 odd nodes
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0.9
Recall the result about traversability
For a network to be traversable it must have 0
or 2 odd nodes and if we are able to start and
finish at the same node it must have no odd
nodes.
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0.9
Recall the result about traversability
For a network to be traversable it must have 0
or 2 odd nodes and if we are able to start and
finish at the same node it must have no odd
nodes.
And consider its implications for the postman
problem.
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0.10
If there are no odd nodes in the network, the
network is certainly traversable.
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0.10
If there are no odd nodes in the network, the
network is certainly traversable.
The minimum distance is the sum of the arc
distances and any Eulerian trail is an
acceptable shortest route, so the problem is
trivial.
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0.11
We now consider the general situation where
there are an even number of odd nodes.
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0.11
We now consider the general situation where
there are an even number of odd nodes.
In such a case the network can be made
traversable by linking together pairs of odd
nodes with additional arcs.
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0.11
We now consider the general situation where
there are an even number of odd nodes.
In such a case the network can be made
traversable by linking together pairs of odd
nodes with additional arcs. The effect of adding
these extra arcs is to make all nodes even and
hence the network becomes traversable.
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0.11
We now consider the general situation where
there are an even number of odd nodes.
In such a case the network can be made
traversable by linking together pairs of odd
nodes with additional arcs. The effect of adding
these extra arcs is to make all nodes even and
hence the network becomes traversable.
Note that this is always possible because the
number of odd nodes is always even.
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0.12
Example
5
104 10
8
6
118
411 4
12
8
ZB
C
A
D G
E F
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0.12
Example
5
104 10
8
6
118
411 4
12
8
ZB
C
A
D G
E F
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0.12
Example
5
104 10
8
6
118
411 4
12
8
ZB
C
A
D G
E F
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0.13
The length of the shortest possible route is the
sum of all the arc lengths
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0.13
The length of the shortest possible route is the
sum of all the arc lengths plus the lengths of
the additional arcs (ZB, AE, DF ).
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0.13
The length of the shortest possible route is the
sum of all the arc lengths plus the lengths of
the additional arcs (ZB, AE, DF ).
Shortest route = 101 + 5 + 4 + 4
= 114
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0.13
The length of the shortest possible route is the
sum of all the arc lengths plus the lengths of
the additional arcs (ZB, AE, DF ).
Shortest route = 101 + 5 + 4 + 4
= 114
There are many possible routes around the
network.
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0.13
The length of the shortest possible route is the
sum of all the arc lengths plus the lengths of
the additional arcs (ZB, AE, DF ).
Shortest route = 101 + 5 + 4 + 4
= 114
There are many possible routes around the
network. Since all nodes are of even order the
postman can start at any node and he finishes
there.
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0.14
The additional arcs to join pairs of odd nodes
were chosen to minimise the additional
distance required to make the network
traversable.
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0.14
The additional arcs to join pairs of odd nodes
were chosen to minimise the additional
distance required to make the network
traversable.
In the example this was done by inspection
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0.14
The additional arcs to join pairs of odd nodes
were chosen to minimise the additional
distance required to make the network
traversable.
In the example this was done by inspection but
in more complex examples an algorithm is
required.
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0.14
The additional arcs to join pairs of odd nodes
were chosen to minimise the additional
distance required to make the network
traversable.
In the example this was done by inspection but
in more complex examples an algorithm is
required.
What algorithm can be used to find a shortest
path?
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0.14
The additional arcs to join pairs of odd nodes
were chosen to minimise the additional
distance required to make the network
traversable.
In the example this was done by inspection but
in more complex examples an algorithm is
required.
What algorithm can be used to find a shortest
path? Dijkstra’s algorithm.
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0.15
It should also be noted that it is not sufficient to
only consider one set of pairings of odd nodes,
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0.15
It should also be noted that it is not sufficient to
only consider one set of pairings of odd nodes,
all such parings must be considered.
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0.15
It should also be noted that it is not sufficient to
only consider one set of pairings of odd nodes,
all such parings must be considered.
So we have the Chinese Postman Algorithm...
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0.16
• Find all nodes of odd order
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0.16
• Find all nodes of odd order
• For every pair of odd nodes find the
connecting path of minimum weight
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0.16
• Find all nodes of odd order
• For every pair of odd nodes find the
connecting path of minimum weight
• Pair up all the odd nodes so that the sum of
weights of connecting paths is minimised
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0.16
• Find all nodes of odd order
• For every pair of odd nodes find the
connecting path of minimum weight
• Pair up all the odd nodes so that the sum of
weights of connecting paths is minimised
• Duplicate the minimum weight paths
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0.16
• Find all nodes of odd order
• For every pair of odd nodes find the
connecting path of minimum weight
• Pair up all the odd nodes so that the sum of
weights of connecting paths is minimised
• Duplicate the minimum weight paths
• Find a trail containing every arc for the new
(Eulerian) graph.
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0.17
• Work through example 5.2.1 (page 59) and
example 5.2.2 of OCR D1
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0.17
• Work through example 5.2.1 (page 59) and
example 5.2.2 of OCR D1
• Practise exercise 5 (page 61) - Yr 12.
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0.17
• Work through example 5.2.1 (page 59) and
example 5.2.2 of OCR D1
• Practise exercise 5 (page 61) - Yr 12.
• Practise exercise 3C (page 93) - Yr 13.
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0.18
Go to..
mymathsteacher.comfor more maths help.
http://www.mymathsteacher.comPresentation produced using LATEX and the Utopia Presentation Bundle
A. G. Martin. December 31, 2001